Electrostatic potential Poisson-Boltzmann equation

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

Calculation of Electrostatic Potential by the Poisson-Boltzmann Equation... 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

For small deviations from electroneutrality, the charge density at x is proportional to -(x)/kT9 where < is the difference of the electrostatic potential from its (constant) value when there is no charge density (the density of a species of charge z is proportional to 1 - zkT on linearizing the Boltzmann exponential). Then the Poisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation ... 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

The Poisson-Boltzmann equation. The slab model is based on a solution of the linearized Poisson-Boltzmann equation that is valid only for low electrostatic surface potentials. As... 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

Figure 6.10 Electrostatic double-layer force between a sphere of R = 3 /um radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson-Boltzmann equation and the Derjaguin approximation for constant potentials (tpi = 80 mV, ip2 = 50 mV) and for constant surface charge (i/2/Ad so that at large distances both lead to the same potential.

The ionic groups on the micellar surface and the counterions will give rise to a nonuniform electrostatic potential according to the Poisson equation. If furthermore the electrostatic effects dominate the counterion distribution the ion concentration is determined by following a Boltzmann distribution. These approximations lead to the Poisson-Boltzmann equation. 

Unlike the other examples in this section, the equation governing the electrostatics here [i.e., Eq. (53)] is not the linearized Poisson-Boltzmann equation. However, considering interactions outside of thin double layers does have the effect of linearizing the problem. In Eq. (54), n is the fluid viscosity, K is the conductivity, is the zeta potential of the z th surface, and is a bipolar coordinate that is constant on the sphere and wall surfaces. It is this last condition (54), derived by Bike and Prieve [36] as a requirement to satisfy charge conservation, that couples the fluid mechanics with the electrostatics. 

The electrostatic field in the stationary state is described by the Poisson-Boltzmann equation. The PB model constitutes the fundamental equation of electrostatics and is based on the differential Poisson equation which describes the electrostatic potential 4>(r) in a medium with a charge density p(r) and a dielectric scalar field e(r) ... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

A thorough discussion of the basic theory describing electrostatic interactions can be found in [7] the pertinent points are discussed below. Electrostatic forces arise from the osmotic pressure difference between two charged surfaces as a result of the local increase in the ionic distribution around each charged surface. For a single electrified interface, the local ion distribution is coupled to the potential distribution near that surface and can be described using the Poisson-Boltzmann equation. The solution of this equation shows that for low surface potentials the potential follows an exponential function with distance from the interface, D, given by... 

In complementary computational studies, Gunner et al. have explored the role of long-range electrostatic interaction on electron transfer processes in the Rhodo-bacter sphaeroides reaction center [38]. The interaction domains were identified by mapping electrostatic potentials, calculated from the Poisson-Boltzmann equation, on to calculated encounter surfaces for each of the components of the reaction center. From qualitative correlation of electron transfer processes with these low-resolution potential maps, it is apparent that long-range interactions profoundly affect the reduction potential of the cofactors in the reaction center. 

Electrostatic interactions in solutions containing charged particles and ions can be described using the Poisson-Boltzmann equation. A charged surface attracts counterions into a double layer of thickness defined by the Debye length, which depends on counterion concentration and solvent dielectric constant. From simplified theories, expressions can be derived for the attractive interaction potential between charged spheres. 

The ion and electrical potential distributions in the electrical double layer can be determined by solving the Poisson-Boltzmann equation [2,3]. According to the theory of electrostatics, the relationship between the eleetrieal potential ij/ and the local net charge density per unit volume at any point in the solution is deseribed by the Poisson equation ... 

Potential energy descriptors proposed as an indicator of hydrophobicity [Oprea and Waller, 1997]. Originally, they were calculated using the finite difference approximation method the linearized Poisson-Boltzmann equation was solved numerically to compute the electrostatic contribution to solvation at each grid point. Desolvation energy field values were calculated as the difference between solvated (grid dielectric = 80) and in vacuo (grid dielectric = 1). 

Dependencies of AG on the average separation l between SC>3 groups and on the distance a between the plane of proton transfer and the negatively charged interface were studied theoretically in Ref. 43, 44. Using an approach based on the Poisson-Boltzmann equation, modulations of the electrostatic potential and of the distribution of mobile protons in the proximity of the charged anionic sites were calculated. 

Equation 5.178 demonstrates that for two identically charged surfaces n, is always positive, i.e., corresponds to repulsion between the surfaces. In general, we have 0 < m < 1, because the coions are repelled from the film due to the interaction with the film surfaces. To find the exact dependence of riel oil tho film thickness, h, we solve the Poisson-Boltzmann equation for the distribution of the electrostatic potential inside the film. The solution provides the following connection between riel 2nd h for symmetric electrolytes " i ... 

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